I know this is probably really easy, but I am having the hardest time understanding what I am supposed to be doing and how I am supposed to be doing it.
It says, for each pair of propositions P & Q, state whether P & Q are logically equivalent.
P= p^q, Q= ~p v~q
Ok, so i don't know really where to start, I know I draw a truth table and have p q on the left side and I have TT, TF, FT, FF....then on the right side I don't know what to do.
Any help, please?
ok, I see it now...and for the final answer you would get P is not equal to Q. right?
I have another question, too.
Another problem says, P= p ^ (q v r), Q= (p v q) ^ (p v r)
so does the table have p, q, r columns to start off with, leaving 8 possible T/F combinations?
dude15129:
So far you know that two statements P AND Q are logically equivalent if they have the same truth tables i.e their true-false values are the same.
I have a question to ask you and anybody else who would care to answer.
You know i believe the h.school theorem :
for all x,y real Nos [absvaleu(x)<y <====> -y<x<y] i.e the two statements
P[=absvalue(x)] and Q[= -y<x<y] are logically equivalent.
How in this case would you test their logical equivalence???
..................SO when we say two statements P AND Q,are logically.....
....................equivalent, if they have the same true tables..........
..................is not generally .................................................. .........
...........................true?.................. .............................................
...................What is generally true then for logical equivalence????.
...The two statements in our case P[=absvalue(x)<y ] and Q[= -y<x<y] are logically equivalent
the statements P and Q themselves are not logical statements that we can array in a truth table. there is simply too many things to check truth values for. you would have to do it for all real numbers x and y, which is insanity, not logic.
the subtle difference here is between syntax and semantics. here we have a syntactic logical equivalence where you are trying to prove it to be a semantic one. it just won't work. not by truth tables. see here
and no, logical statements and equaivalences are not always "true" in general. we can set up definitions to say whatever we want to say
There is not syntactic or semantic logical equivalence,but we can prove syntactically or semantically the logical equivalence.
now let see you give a syntactical proof of the following equivalences:
1) p^(qvr) <===> (p^q)v(p^r)
2) (x)(y)[ absvalue(x)<y<====> -y<x<y], where (x),(y) means for all x,y real Nos
since you claim to know the stuff it will be a good paradigm for those learning logic how to apply it in mathematics.
Now the proof must be in steps and every step must be justified